Bayes' rule: posterior ∝ prior × likelihood. The prior is your belief before any data; the likelihood is how compatible each θ is with what you flipped (heads h, tails t). Multiply, normalise, and you have the posterior — your updated belief.
The prior here is a Beta distribution. Beta(α, β) is conjugate to coin flips, so the posterior after h heads and t tails is just Beta(α + h, β + t). From your sliders, α = mean · strength and β = (1 − mean) · strength. The posterior mean is α/(α+β).